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Sharpening the Becker-Stark Inequalities
Journal of Inequalities and Applications volume 2010, Article number: 931275 (2010)
Abstract
In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.
1. Introduction
Steckin [1] (or see Mitrinovic [2, 3.4.19, page 246]) gives us a result as follows.
Theorem 1.1 (see [1, Lemma ]).
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ1_HTML.gif)
Later, Becker and Stark [3] (or see Kuang [4, 5.1.102, page 248]) obtain the following two-sided rational approximation for .
Theorem 1.2.
Let , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ2_HTML.gif)
Furthermore, and
are the best constants in (1.2).
In fact, we can obtain the following further results.
Theorem 1.3.
Let , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ3_HTML.gif)
Furthermore, and
are the best constants in (1.3).
In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.
Theorem 1.4.
Let , and let
be a natural number. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ4_HTML.gif)
holds, where , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ5_HTML.gif)
where are the even-indexed Bernoulli numbers.
Furthermore, and
are the best constants in (1.4).
2. Four Lemmas
Lemma 2.1.
The function is decreasing, where
is Riemann's zeta function.
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_IEq18_HTML.gif)
is equivalent to the function , which is decreasing.
Lemma 2.2 (see [5, Theorem ]).
Let be Riemann's zeta function and
the even-indexed Bernoulli numbers. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ6_HTML.gif)
Lemma 2.3 (see [6, 1.3.1.4 (1.3)]).
Let . Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ7_HTML.gif)
Lemma 2.4.
Let and
. Then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ8_HTML.gif)
Proof.
By Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ9_HTML.gif)
Since is decreasing by Lemma 2.1, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ10_HTML.gif)
From Lemma 2.2, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ11_HTML.gif)
which implies that for
.
3. Proofs of Theorems
Proof of Theorem 1.4.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ12_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F931275/MediaObjects/13660_2009_Article_2303_Equ13_HTML.gif)
By Lemma 2.4, we have for
, and
is decreasing on
.
At the same time, =
by (3.1), and
by (3.2), so
and
are the best constants in (1.4).
Proof of Theorem 1.3.
Let in Theorem 1.4; we obtain that
and
. Then the proof of Theorem 1.3 is complete.
References
Steckin SB: Some remarks on trigonometric polynomials. Uspekhi Matematicheskikh Nauk 1955, 10(1(63)):159–166.
Mitrinovic DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.
Becker M, Strak EL: On a hierarchy of quolynomial inequalities for tanx. University of Beograd Publikacije Elektrotehnicki Fakultet. Serija Matematika i fizika 1978, (602–633):133–138.
Kuang JC: Applied Inequalities. 3rd edition. Shangdong Science and Technology Press, Jinan City, China; 2004.
Scharlau W, Opolka H: From Fermat to Minkowski, Undergraduate Texts in Mathematics. Springer, New York, NY, USA; 1985:xi+184.
Jeffrey A: Handbook of Mathematical Formulas and Integrals. 3rd edition. Elsevier Academic Press, San Diego, Calif, USA; 2004:xxvi+453.
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Zhu, L., Hua, J. Sharpening the Becker-Stark Inequalities. J Inequal Appl 2010, 931275 (2010). https://doi.org/10.1155/2010/931275
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DOI: https://doi.org/10.1155/2010/931275